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In mathematics, the Radon–Nikodym theorem is a result in measure theory which states that, given a measurable space , if a σ-finite measure on is absolutely continuous with respect to a σ-finite measure on , then there is a measurable function , such that for any measurable subset : : The function is called the Radon–Nikodym derivative and denoted by . The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is in 1913, and for Otto Nikodym who proved the general case in 1930. In 1936 Hans Freudenthal further generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory, which contains the Radon–Nikodym theorem as a special case. If is a Banach space and the generalization of the Radon–Nikodym theorem also holds for functions with values in (mutatis mutandis), then is said to have the Radon–Nikodym property. All Hilbert spaces have the Radon–Nikodym property. ==Radon–Nikodym derivative== The function satisfying the above equality is ''uniquely defined up to a -null set'', that is, if is another function which satisfies the same property, then -almost everywhere. is commonly written and is called the Radon–Nikodym derivative. The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another (the way the Jacobian determinant is used in multivariable integration). A similar theorem can be proven for signed and complex measures: namely, that if is a nonnegative σ-finite measure, and ''ν'' is a finite-valued signed or complex measure such that ''ν ≪ μ'', i.e. ''ν'' is absolutely continuous with respect to , then there is a -integrable real- or complex-valued function on such that for every measurable set , : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Radon–Nikodym theorem」の詳細全文を読む スポンサード リンク
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